Subgroups — #28298862

Order 13568.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289822	C₂	1	2	yes	maximal
2	#20272631	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133267	C₄	1	4	yes	—
4	C₄×#11719908	C₂²	1	1	yes	characteristic
5	C₈×#88322	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	#13133267	C₈	1	8	yes	complemented
8	C₂×#11719908	C₂×C₄	1	2	yes	—
9	C₄×#88322	C₈	1	2	yes	—
10	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
11	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
14	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
15	C₂×#88322	—	2	2	no	—
16	C₁₆×C₅₃	—	2	2	no	—
17	#11719908	C₂×C₈	1	4	yes	—
18	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
19	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
20	C₈×C₅₃	C₄×C₈	1	2	yes	—
21	C₈×C₅₃	M₃₂	1	2	yes	—
22	#88322	—	4	2	no	—
23	#7330	—	53	1	no	maximal, Sylow
24	C₂²×C₅₃	#346	1	1	yes	characteristic
25	C₄×C₅₃	—	2	2	no	—
26	C₄×C₅₃	C₈²	1	1	yes	characteristic
27	C₄×C₅₃	#346	1	1	yes	characteristic
28	#1397	—	53	2	no	—
29	#1311	—	53	1	no	—
30	C₂×C₅₃	#1311	1	1	yes	characteristic
31	C₂×C₅₃	#1398	1	2	yes	—
32	C₂×C₃₂	—	53	4	no	—
33	C₄×C₁₆	—	53	1	no	—
34	C₄×C₁₆	#6157	1	1	yes	characteristic
35	C₈²	—	53	1	no	—
36	C₅₃	#7330	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₁₆	—	53	2	no	—
38	C₂×C₁₆	#88324	1	2	yes	—
39	C₄×C₈	—	53	2	no	—
40	C₄×C₈	C₂×#6157	1	1	yes	characteristic
41	C₃₂	—	53	8	no	—
42	C₂×C₈	—	106	2	no	—
43	C₂×C₈	C₄×#6157	1	1	yes	characteristic
44	C₂×C₈	C₂×#88324	1	1	yes	characteristic
45	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
46	C₁₆	—	2	2	no	—
47	C₁₆	—	53	4	no	—
48	C₂×C₄	C₄×#88324	1	1	yes	characteristic
49	C₂×C₄	#13133292	1	2	yes	—
50	C₈	—	212	2	no	—
51	C₈	C₈×#6157	1	2	yes	—
52	C₈	#13133291	1	2	yes	—
53	C₂²	#16811512	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	C₈×#88324	1	1	yes	characteristic
56	C₄	#16811511	1	1	yes	characteristic
57	C₂	#20289627	1	1	yes	characteristic
58	C₂	#20289827	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×#6157
commutator	C₄×C₅₃	C₈²
Frattini	C₄×C₈	C₂×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	#346
cosocle	C₄×#11719908	C₂²
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#7330
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. #13133267	C₈	semidirect
36. C₅₃	#7330	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20289822	2
#20272631	2
#7330	53